Topic: Consistency proof

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In the News (Wed 19 Dec 18)

M. E. Szabo: The Collected Works of Gerhard Gentzen(Site not responding. Last check: 2007-10-20)

After developing and examining the 'natural calculus' Gentzen conjectured that proofs should have a "normal form" in which "all concepts required for the proof would in some sense appear in the conclusion of the proof." This is the subformula property.

Consistency follows from the fact that the sequent "1 = 2" is not in reduced form but any derivation of the sequent cannot be reduced.

In #8 [1938], Gentzen gives a new proof of consistency of elementary number theory based on his sequent calculus LK (where the one in #4 was based on natural deduction).

www.andrew.cmu.edu /user/cebrown/notes/szabo.html (3057 words)

Reflections(Site not responding. Last check: 2007-10-20)

The symposium is centered around proof theoretically inspired foundational investigations that have been merging over the last decades with developments in set theory and recursion theory; however, they have sustained a special emphasis on broad philosophical issues.

The most distinctive aspect of Gentzen's consistencyproof for elementary arithmetic was the use of quantifier-free transfinite induction up to the first epsilon number, epsilon0.

The attempt to establish the consistency of stronger and stronger classical theories was accompanied by the systematic development of analysis in weaker and weaker formal theories.

www-formal.stanford.edu /clt/Reflections/symposium.html (1011 words)

Hilbert's Program(Site not responding. Last check: 2007-10-20)

Proof theory in the tradition of Gentzen has analyzed axiomatic systems according to what extensions of the finitary standpoint are necessary to prove their consistency.

What is required for a consistencyproof is an operation which, given a formal derivation, transforms such a derivation into one of a special form, plus proofs that the operation in fact does this and that proofs of the special kind cannot be proofs of an inconsistency.

Proof theorists have obtained a number of such results, including reductions of theories which on their face require a significant amount of ideal mathematics for their justification (e.g., subsystems of analysis) to finitary systems.

Part 3: A Very Very Very Very Very Very Pathetic and Ignorant Book(Site not responding. Last check: 2007-10-20)

This is an example of a relative consistencyproof, which reduces the consistencyproof of one system to that of another.

This implies to me that all known consistencyproofs of arithmetic rely on something like transfinite induction (or possibly primitive recursive functionals of finite type), the consistency of which is no more self- evident than that of arithmetic itself.

For example, Gentzen's proof of the consistency of PA uses transfinite induction.

www.mathpages.com /home/kmath347/kmvs03.htm (965 words)

From Frege To Godel: von Heijenoort(Site not responding. Last check: 2007-10-20)

The proof was controversial, and Zermelo later discussed the reactions and presented a second proof.

The fact that Konig had a [flawed] proof that the continuum cannot be well-ordered coupled with the fact that Zermelo had proven that all sets can be well-ordered appeared to imply that set theory contained contradictions stemming from the notion of well-ordering.

Zermelo also discusses objections to the first proof which included a mistrust of Cantor's set theory, a wariness of the principle of choice, and a suspicion that the argument is reminiscent of those leading to the paradoxes.

www.andrew.cmu.edu /~cebrown/notes/vonHeijenoort.html (8419 words)

Proof Theory on the eve of Year 2000(Site not responding. Last check: 2007-10-20)

Add to this that it is closely connected with the proof theory of feasible arithmetic, and it seems clear to me that it is a classic problem of proof theory, though one that was quite unconsidered by the early pioneers.

A precise representation of mathematical proofs as formal(izable) derivations is sought.

Proof theorists, having failed in analysing proofs in mathematics, went on to apply their skills (somewhat opportunistically in my mind) in logical systems different from the two canonical ones, intuitionistic and classical.

www-logic.stanford.edu /proofsurvey.html (19489 words)

Proof Theoretic Strength(Site not responding. Last check: 2007-10-20)

Variant (b) concerns what is required to prove the consistency of the system, and is the least well-ordering necessary for that purpose.

Under this account of Proof Theoretic Strength one formal system is at strictly stronger than another if the consistency of the second can be proven in the first.

A more comprehensive discussion is in Pohlers' book Proof Theory, which is basically about ordinal analysis, which means discovering the proof theoretic (or ordinal) strength of axiom systems.

Gentzen's consistencyproofs for arithmetic launched a field of research known as ordinal analysis, and the program of measuring the strength of mathematical theories using ordinal notations is still pursued today.

Godel's proof is not easy to follow, nor easy to grasp the full implications of its conclusions.

In their short book (118 pages) Nagel and Newman present the basic structure of Godel's proof and the core of his conclusions in a way that is intelligible to the persistent layman.

Following chapters explain Hilbert's approach to the consistency problem - the formalization of a deductive system, the meaning of model-based consistency versus absolute consistency, and gives an example of a successful absolute proof of consistency.

www.thebusinessbookstore.com /Books/Godels_Proof.html (798 words)

Homage to Kurt Godel(Site not responding. Last check: 2007-10-20)

Consequently, any proof of consistency of a logical system (from within that system) is a proof of Gödel's fork in that system, which implies inconsistency in the system.

Inescapably, since M believes that L contains a proof of Gödel's fork in L, and M contains all the axioms and methods of reasoning open to L, M is able to deduce that L is inconsistent.

Proofs get picked up from the left (by imply and forbid) while disproofs are gathered from the right (by unless and demands); according as what is produced is a Note that

In his proof of A7, Hunter leaves the proof of an important lemma to the reader, namely, that the set of all strings of 1's and 0's that consist of all 0's after a certain point are enumerable.

In Post's model-theoretic proof of the consistency of PS, we looked for and found a single interpretation of P that was also a model.

The strange 'truth table' constructed in the proof of metatheorem 36.1 helps us contrive a strange interpretation I such that axiom-schema #1 is false for I while the other two axiom-schemata of PS are true for I. But this I is so strange that, e.g.

www.earlham.edu /~peters/courses/logsys/exercise.htm (6908 words)

Higher-Order Proof by Consistency (ResearchIndex)(Site not responding. Last check: 2007-10-20)

Abstract: Weinvestigate an integration of the first-order method of proof by consistency (PBC), also known as term rewriting induction, into theorem proving in higher-order specifications.

How is Gödel's theorem relevant in criticizing such a view of the axioms and methods of proof of T? Note that I'm not arguing here that we do have absolute knowledge in mathematics, or that we are justified in accepting the methods and axioms of T without proof.

Now Gödel's theorem implies that the consistency of T cannot be proved in T. Why should this be an argument against our accepting the axioms and methods of T? After all, we already know that not everything in mathematics can be mathematically proved.

Logically, it is perfectly compatible with "T is consistent" that (i) "there are infinitely many primes in the series 3,8,13,18..." is provable in T, and (ii) there are only finitely many primes in the series 3,8,13,18....

Its conclusions are as strange as they are profound, but, unlike other recent theorems of comparable importance, grasping the main steps of the proof requires little more than high school algebra and a bit of patience.

Where the first edition fell down, however, was in the guts of the proof itself; the brevity that served so well in defining the problem made their rendering of Gödel's solution so dense as to be nearly indigestible.

Marking the 70th anniversary of the original publication of Gödel's Proof, New York University Press is proud to publish this special anniversary edition of one of its bestselling books.